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The setup is similar to quantum eraser experiments (see below). The laser pulses at regular intervals to send bunches of photons. They get split in half as entangled particles. One beam is sent 1 light year in one direction to (Z) and the other sent 1 light year in the other direction to the set of detectors (A,B,C,D). If the mirrors (X & Y) are in place at the detectors and they cause A & B to measure which slit the photons pass through then they will collapse the corresponding entangled particles at Z. An observer at Z will not see an interference pattern for that bunch of photons. However if the mirrors (X & Y) are removed then the observer at Z will see the entangled bunch of photons form an interference pattern. It seems like you could then use whether each incoming bunch of photons forms an interference pattern or not as binary. Obviously this setup doesn't work because FTL communication is impossible, I'm trying to understand why it wouldn't work.

Note in response to the first answer: Sending entangled light particles a light year to a detector is of course difficult or impossible. I chose an arbitrarily long length to exaggerate its effect. The same question could be posed with a light beam of any length.

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I am not entirely sure I get your setup, since you have separate beams but they seem to emerge from the same double slit. Perhaps someone who gets the setup well can have a deeper answer for this, but there are right of the bat several practical reasons why this cannot work.

You are building an interstellar interferometer. For you to see wave-like phenomena you need to maintain coherence over the entire interferometer length. While coherence length for ultra-stable oscillators can be interplenatry, the linewidth you would need for your scenario or anything approaching it is not practical. Let alone that you have to align the interferometer somehow, and there still has to be spatial overlap while any beam of radiation will inevitably spread through diffraction making your capture efficiency below practical. And your beams are not going to be enormous, well-collimated beams either because as per setup they just passed a double-slit and will be diffracting heavily.